---
tag: 0T3GK0XCBC
kind: concept
name: Phase portrait
msc: [34C05, 37C10]
uses: [05VDGH129F, 0BVT6VSB3S, 0G4X36AE0X]
sources:
  - url: https://en.wikipedia.org/wiki/Phase_portrait
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Phase_portrait
    locator: Visualizing the behavior of ordinary differential equations
same_as:
  wikidata: Q2532943
  wikipedia: Phase portrait
---

## Summary

A phase portrait of an autonomous system is a picture of a representative set of its [[trajectories|0BVT6VSB3S]] in the phase space, usually the [[phase plane|05VDGH129F]].
Arrows on the curves show the direction of time.
It shows the [[equilibrium points|0G4X36AE0X]], the periodic orbits and the long-time behaviour of the solutions.

## Remarks

The [[phase line|081QSEAGEY]] is the phase portrait of a scalar autonomous equation.
For the undamped [[harmonic oscillator|0YG0ZJPTWG]] $x'' = -x$, the phase portrait in the $(x, x')$ plane consists of the origin and the circles around it.

## Explanations

### L1 human

A phase portrait is a map of all the possible motions of a system.
One look shows where the system rests, where it cycles and where it goes in the long run.